Calculation of Laminar Flow through Fracture

The calculation of laminar flow through fracture is very fundamental (), and it is very important for my research, also. Late last year, I derived the equation for both Newtonian fluid and non-Newtonian fluid by hand. Now I am going to make a note for that derivation for Newtonian fluid, which could be regarded as the answer to an exercise on Transport Phenomena, 2nd Ed. by Bird, R.B., et al., a classical textbook on this topic.

At first we need to describe the physical properties of the fracture model (as shown in Figure.1).

Figure 1. Physical Properties of Fracture Model

A Newtonian fluid is in laminar flow in a narrow slit formed by two parallel walls with length, L, a distance B (fracture width) apart. It is understood that the fracture height, B《W, so that “edge effects” are unimportant. Make a differential momentum balance, and obtain the following expressions for the momentum-flux and velocity distributions: